A Petrov???Galerkin reduced basis approximation of the Stokes equation in parameterized geometries, Comptes Rendus Mathematique, vol.353, issue.7, pp.641-645, 2015. ,
DOI : 10.1016/j.crma.2015.03.019
Abstract, Computational Methods in Applied Mathematics, vol.15, issue.2 ,
DOI : 10.1515/cmam-2015-0004
Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs, 2000. ,
On the existence and regularity of the solution of Stokes problem in arbitrary dimension, Proc. Japan. Acad, pp.171-175, 1991. ,
DOI : 10.3792/pjaa.67.171
$hp$-Version Composite Discontinuous Galerkin Methods for Elliptic Problems on Complicated Domains, SIAM Journal on Scientific Computing, vol.35, issue.3 ,
DOI : 10.1137/120877246
On the implementation of mixed methods as nonconforming methods for second-order elliptic problems, Math. Comp, vol.64, pp.943-972, 1995. ,
Mixed finite element methods for elliptic problems, Computer Methods in Applied Mechanics and Engineering, vol.82, issue.1-3 ,
DOI : 10.1016/0045-7825(90)90168-L
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.165.5348
Mixed and nonconforming finite element methods : implementation, postprocessing and error estimates, ESAIM: Mathematical Modelling and Numerical Analysis, vol.19, issue.1 ,
DOI : 10.1007/BF01396493
URL : http://www.esaim-m2an.org/articles/m2an/pdf/1985/01/m2an1985190100071.pdf
Analyse fonctionnelle appliquée, 1987. ,
The nonconforming virtual element method, ESAIM: Mathematical Modelling and Numerical Analysis, vol.50, issue.3, 2015. ,
DOI : 10.1016/B978-0-12-068650-6.50030-7
The $h-p$ version of the finite element method with quasiuniform meshes, ESAIM: Mathematical Modelling and Numerical Analysis, vol.21, issue.2, pp.199-238, 1987. ,
DOI : 10.1007/978-3-642-66451-9
-Version of the Finite Element Method, SIAM Journal on Numerical Analysis, vol.24, issue.4, pp.750-776, 1987. ,
DOI : 10.1137/0724049
-Version of the Finite Element Method, SIAM Journal on Numerical Analysis, vol.18, issue.3, pp.515-545, 1981. ,
DOI : 10.1137/0718033
The finite element method with Lagrangian multipliers, Numerische Mathematik, vol.12, issue.3, pp.179-192, 1973. ,
DOI : 10.1090/trans2/057/08
An ???empirical interpolation??? method: application to efficient reduced-basis discretization of partial differential equations, Comptes Rendus Mathematique, vol.339, issue.9, pp.667-672, 2004. ,
DOI : 10.1016/j.crma.2004.08.006
URL : https://hal.archives-ouvertes.fr/hal-00021702
On the flexibility of agglomeration based physical space discontinuous Galerkin discretizations, Journal of Computational Physics, vol.231, issue.1, pp.45-65, 2012. ,
DOI : 10.1016/j.jcp.2011.08.018
URL : https://hal.archives-ouvertes.fr/hal-00562219
A Note on the Poincar?? Inequality for Convex Domains, Zeitschrift f??r Analysis und ihre Anwendungen ,
DOI : 10.4171/ZAA/1170
BASIC PRINCIPLES OF VIRTUAL ELEMENT METHODS, Mathematical Models and Methods in Applied Sciences, vol.61, issue.01, pp.199199-214, 2013. ,
DOI : 10.1051/m2an:2008030
Basic principles of hp virtual elements on quasiuniform meshes, Mathematical Models and Methods in Applied Sciences, vol.2, issue.08, pp.1567-1598, 2016. ,
DOI : 10.1142/S0218202515500372
Mimetic finite difference method for the Stokes problem on polygonal meshes, Journal of Computational Physics, vol.228, issue.19, pp.2287215-7232, 2009. ,
DOI : 10.1016/j.jcp.2009.06.034
Convergence rates for Greedy algorithms in Reduced?Basis methods, SIAM J. Math Anal, 2011. ,
DOI : 10.21236/ada640047
URL : https://hal.archives-ouvertes.fr/hal-00767082
A Nonconforming High-Order Method for the Biot Problem on General Meshes, SIAM Journal on Scientific Computing, vol.38, issue.3, pp.1508-1537, 2016. ,
DOI : 10.1137/15M1025505
URL : https://hal.archives-ouvertes.fr/hal-01162976
Mixed finite element methods and applications, 2013. ,
DOI : 10.1007/978-3-642-36519-5
Unified formulation and analysis of mixed and primal discontinuous skeletal methods on polytopal meshes, ESAIM: Mathematical Modelling and Numerical Analysis, 2016. ,
DOI : 10.1051/m2an/2017036
URL : https://hal.archives-ouvertes.fr/hal-01365938
Analysis of Compatible Discrete Operator Schemes on polyhedral meshes for elliptic and Stokes equations, 2014. ,
Analysis of compatible discrete operator schemes for Stokes problems on polyhedral meshes, IMA J. Numer. Anal, vol.34, issue.4, pp.553-581, 2014. ,
Reduced-Basis Approach for Homogenization beyond the Periodic Setting, Multiscale Modeling & Simulation, vol.7, issue.1, pp.466-494, 2008. ,
DOI : 10.1137/070688791
URL : https://hal.archives-ouvertes.fr/inria-00132763
Reduced Basis Techniques for Stochastic Problems, Archives of Computational Methods in Engineering, vol.8, issue.1, pp.435-454, 2010. ,
DOI : 10.1051/cocv:2002041
URL : https://hal.archives-ouvertes.fr/hal-00470522
A reduced basis approach for variational problems with stochastic parameters: Application to heat conduction with variable Robin coefficient, Computer Methods in Applied Mechanics and Engineering, vol.198, issue.41-44, pp.41-443187, 2009. ,
DOI : 10.1016/j.cma.2009.05.019
URL : https://hal.archives-ouvertes.fr/inria-00311463
On the existence, uniqueness and approximation of saddle-point problems arising from lagrange multipliers, RAIRO Ser. Rouge, vol.8, pp.129-151, 1874. ,
Mixed and hybrid finite element methods, 2013. ,
DOI : 10.1007/978-1-4612-3172-1
Convergence of the Mimetic Finite Difference Method for Diffusion Problems on Polyhedral Meshes, SIAM Journal on Numerical Analysis, vol.43, issue.5 ,
DOI : 10.1137/040613950
A priori convergence of the Greedy algorithm for the parametrized Reduced-Basis method, ESAIM: M2AN, pp.595-603, 2012. ,
URL : https://hal.archives-ouvertes.fr/hal-00659314
BUBBLE STABILIZED DISCONTINUOUS GALERKIN METHOD FOR STOKES' PROBLEM, Mathematical Models and Methods in Applied Sciences, vol.9, issue.02, pp.297-313, 2010. ,
DOI : 10.1142/S0218202502002240
URL : https://hal.archives-ouvertes.fr/hal-01090930
hp-Version discontinuous Galerkin methods on polygonal and polyhedral meshes, Mathematical Models and Methods in Applied Sciences, vol.84, issue.10, pp.2009-2041, 2014. ,
DOI : 10.1016/S0898-1221(03)90088-5
Optimal a priori error estimates for the $hp$-version of the local discontinuous Galerkin method for convection--diffusion problems, Mathematics of Computation, vol.71, issue.238, pp.71455-478, 2001. ,
DOI : 10.1090/S0025-5718-01-01317-5
Su un problema al contorno relativo al sistema di equazioni di Stokes, Rend. Sem. Mat. Univ. Padova, vol.31, pp.308-340, 1961. ,
Analysis of HDG Methods for Oseen Equations, Journal of Scientific Computing, vol.35, issue.3, pp.392-431, 2013. ,
DOI : 10.1007/BFb0064470
Une m??thode combin??e d'??l??ments finis ?? deux grilles/bases r??duites pour l'approximation des solutions d'une E.D.P. param??trique, Comptes Rendus Mathematique, vol.347, issue.7-8, pp.435-440, 2009. ,
DOI : 10.1016/j.crma.2009.02.019
A Hybrid High- Order method for the Cahn?Hilliard problem in mixed form ,
URL : https://hal.archives-ouvertes.fr/hal-01477247
Equivalence between and multigrid algorithms for nonconforming and mixed methods for second-order elliptic problems, East-West J. Numer. Math, vol.4, pp.1-33, 1996. ,
The finite element method for elliptic problems, #25001)], p.520174, 1958. ,
Bridging the hybrid high-order and hybridizable discontinuous Galerkin methods, ESAIM: Mathematical Modelling and Numerical Analysis, vol.50, issue.3, pp.635-650, 2016. ,
DOI : 10.1007/978-3-642-22980-0
URL : https://hal.archives-ouvertes.fr/hal-01115318
The Derivation of Hybridizable Discontinuous Galerkin Methods for Stokes Flow, SIAM Journal on Numerical Analysis, vol.47, issue.2, pp.1092-1125, 2009. ,
DOI : 10.1137/080726653
Unified Hybridization of Discontinuous Galerkin, Mixed, and Continuous Galerkin Methods for Second Order Elliptic Problems, SIAM Journal on Numerical Analysis, vol.47, issue.2, pp.1319-1365, 2009. ,
DOI : 10.1137/070706616
The local discontinuous Galerkin method for the Oseen equations, Mathematics of Computation, vol.73, issue.246, pp.569-593, 2003. ,
DOI : 10.1090/S0025-5718-03-01552-7
Devising methods for Stokes flow: An overview, Computers & Fluids, vol.98, pp.221-229, 2014. ,
DOI : 10.1016/j.compfluid.2013.11.017
Conforming and nonconforming finite element methods for solving the stationary Stokes equations I, Revue fran??aise d'automatique informatique recherche op??rationnelle. Math??matique, vol.7, issue.R3 ,
DOI : 10.1016/B978-0-12-068650-6.50020-4
Displacement and Equilibrium Models in the Finite Element Method, Stress Analysis, pp.145-197, 1977. ,
DOI : 10.1007/978-94-009-9147-7_3
A Supernodal Approach to Sparse Partial Pivoting, SIAM Journal on Matrix Analysis and Applications, vol.20, issue.3, pp.720-755, 1999. ,
DOI : 10.1137/S0895479895291765
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.131.6235
Analysis of a discontinuous Galerkin approximation of the Stokes problem based on an artificial compressibility flux, International Journal for Numerical Methods in Fluids, vol.9, issue.8, pp.55793-813, 2007. ,
DOI : 10.1007/978-1-4757-4355-5
A Hybrid High-Order method for Leray? Lions elliptic equations on general meshes Accepted for publication, Math. Comp, 2016. ,
A Discontinuous-Skeletal Method for Advection-Diffusion-Reaction on General Meshes, SIAM Journal on Numerical Analysis, vol.53, issue.5, pp.2135-2157, 2015. ,
DOI : 10.1137/140993971
URL : https://hal.archives-ouvertes.fr/hal-01079342
Mathematical aspects of discontinuous Galerkin methods, of Mathématiques & Applications ,
DOI : 10.1007/978-3-642-22980-0
A hybrid high-order locking-free method for linear elasticity on general meshes, Computer Methods in Applied Mechanics and Engineering, vol.283, pp.1-21, 2015. ,
DOI : 10.1016/j.cma.2014.09.009
URL : https://hal.archives-ouvertes.fr/hal-00979435
Arbitrary-order mixed methods for heterogeneous anisotropic diffusion on general meshes, IMA Journal of Numerical Analysis, vol.37, issue.1, 2016. ,
DOI : 10.1093/imanum/drw003
URL : https://hal.archives-ouvertes.fr/hal-00918482
Discontinuous Galerkin Methods for Anisotropic Semidefinite Diffusion with Advection, SIAM Journal on Numerical Analysis, vol.46, issue.2, pp.805-831, 2008. ,
DOI : 10.1137/060676106
An arbitrary-order and compactstencil discretization of diffusion on general meshes based on local reconstruction operators, Comput. Meth. Appl. Math, vol.14, issue.4, pp.461-472, 2014. ,
URL : https://hal.archives-ouvertes.fr/hal-00978198
A review of Hybrid High-Order methods: formulations computational aspects, comparison with other methods. Bibliography Building bridges: Connections and challenges in modern approaches to numerical partial differential equations, 2016. ,
URL : https://hal.archives-ouvertes.fr/hal-01163569
A discontinuous skeletal method for the viscosity-dependent Stokes problem, Computer Methods in Applied Mechanics and Engineering, vol.306 ,
DOI : 10.1016/j.cma.2016.03.033
URL : https://hal.archives-ouvertes.fr/hal-01244387
An extension of the Crouzeix???Raviart space to general meshes with application to quasi-incompressible linear elasticity and Stokes flow, Mathematics of Computation, vol.84, issue.291, pp.1-31, 2015. ,
DOI : 10.1090/S0025-5718-2014-02861-5
URL : https://hal.archives-ouvertes.fr/hal-00753660
An a posteriori-driven adaptive Mixed High-Order method with application to electrostatics, Journal of Computational Physics, vol.326, issue.1, pp.35-55, 2016. ,
DOI : 10.1016/j.jcp.2016.08.041
URL : https://hal.archives-ouvertes.fr/hal-01310313
Mixed finite element methods for second order elliptic problems, Math. Appl. Comp, vol.1, pp.91-103, 1982. ,
A mixed finite volume scheme for anisotropic diffusion problems on any grid, Numerische Mathematik, vol.59, issue.1, pp.35-71, 2006. ,
DOI : 10.1007/s00211-006-0034-1
URL : https://hal.archives-ouvertes.fr/hal-00005565
A UNIFIED APPROACH TO MIMETIC FINITE DIFFERENCE, HYBRID FINITE VOLUME AND MIXED FINITE VOLUME METHODS, Mathematical Models and Methods in Applied Sciences, vol.17, issue.02, pp.1-31, 2010. ,
DOI : 10.1007/s10596-004-3771-1
URL : https://hal.archives-ouvertes.fr/hal-00346077
Polynomial approximation of functions in Sobolev spaces, Mathematics of Computation, vol.34, issue.150, pp.441-463, 1980. ,
DOI : 10.1090/S0025-5718-1980-0559195-7
A Hybrid Discontinuous Galerkin method for Darcy? Stokes problems. Domain Decomposition Methods in Science and Engineering XX, 2009. ,
DOI : 10.1007/978-3-642-35275-1_79
Theory and practice of finite elements, Applied Mathematical Sciences, vol.159, 2004. ,
DOI : 10.1007/978-1-4757-4355-5
Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes SUSHI: a scheme using stabilization and hybrid interfaces, IMA Journal of Numerical Analysis, vol.30, issue.4, pp.1009-1043, 2010. ,
DOI : 10.1093/imanum/drn084
A Stream Virtual Element Formulation of the Stokes Problem on Polygonal Meshes, SIAM Journal on Numerical Analysis, vol.52, issue.1 ,
DOI : 10.1137/13091141X
Error estimates for mixed methods, RAIRO Anal. numer, vol.14, pp.309-324, 1980. ,
Optimal error estimates for the hp-version interior penalty discontinuous Galerkin finite element method, IMA Journal of Numerical Analysis, vol.25, issue.1 ,
DOI : 10.1093/imanum/drh014
-Adaptive composite discontinuous Galerkin methods for elliptic problems on complicated domains, Numerical Methods for Partial Differential Equations, vol.200, issue.4, pp.1342-1367, 2014. ,
DOI : 10.1016/j.cma.2011.04.017
URL : https://hal.archives-ouvertes.fr/in2p3-01415342
Finite element methods for Navier-Stokes equations Theory and algorithms, of Springer Series in Computational Mathematics, 1986. ,
A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier-Stokes problems, Mathematics of Computation, vol.74, issue.249, pp.53-84, 2004. ,
DOI : 10.1090/S0025-5718-04-01652-7
URL : https://hal.archives-ouvertes.fr/hal-00020211
Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations, ESAIM: Mathematical Modelling and Numerical Analysis, vol.49, issue.3, pp.575-605, 2007. ,
DOI : 10.1109/8.929635
URL : https://hal.archives-ouvertes.fr/hal-00112154
Reduced basis method for finite volume approximations of parametrized linear evolution equations, ESAIM: Mathematical Modelling and Numerical Analysis, vol.337, issue.2, pp.277-302, 2008. ,
DOI : 10.1016/j.crma.2003.09.023
New development in freefem++, Journal of Numerical Mathematics, vol.20, issue.3-4, pp.251-265, 2012. ,
DOI : 10.1515/jnum-2012-0013
URL : https://hal.archives-ouvertes.fr/hal-01476313
Benchmark on discretization schemes for anisotropic diffusion problems on general grids, Finite Volumes for Complex Applications V, pp.659-692, 2008. ,
URL : https://hal.archives-ouvertes.fr/hal-00429843
Certified Reduced-Basis Methods for Parametrized Differential Equations, SpringerBriefs in Mathematics, 2016. ,
DOI : 10.1007/978-3-319-22470-1
URL : https://hal.archives-ouvertes.fr/hal-01223456
A hybridized finite element method for the Stokes problem, Computers & Mathematics with Applications, vol.68, issue.12, pp.2222-2232, 2014. ,
DOI : 10.1016/j.camwa.2014.08.005
BOOST uBLAS C++ Library ,
A finite volume method for the approximation of highly anisotropic diffusion operators on unstructured meshes, Finite volumes for complex applications IV, pp.401-412, 2005. ,
A high-order mimetic method on unstructured polyhedral meshes for the diffusion equation, Journal of Computational Physics, vol.272, pp.360-385, 2014. ,
DOI : 10.1016/j.jcp.2014.04.021
A weak Galerkin finite element method for the Oseen equations, Advances in Computational Mathematics, vol.15, issue.6, pp.1-18, 2016. ,
DOI : 10.1007/978-1-4612-3172-1
A priori convergence theory for Reduced-Basis approximations of single-parameter elliptic partial differential equations, Journal of Scientific Computing, vol.17, issue.1/4, pp.437-446, 2002. ,
DOI : 10.1023/A:1015145924517
URL : https://hal.archives-ouvertes.fr/hal-00798389
A reduced-basis element method, Comptes Rendus Mathematique, vol.335, issue.2, pp.447-459, 2002. ,
DOI : 10.1016/S1631-073X(02)02427-5
URL : https://hal.archives-ouvertes.fr/hal-00112608
An Inexpensive Method for the Evaluation of the Solution of the Lowest Order Raviart???Thomas Mixed Method, SIAM Journal on Numerical Analysis, vol.22, issue.3, pp.493-496, 1985. ,
DOI : 10.1137/0722029
Discontinuous Galerkin methods for the Stokes equations using divergence???free approximations, International Journal for Numerical Methods in Fluids, vol.27, issue.5 ,
DOI : 10.1007/978-1-4612-3172-1
A hp-analysis of the Local Discontinuous Galerkin method for diffusion problems, J. Sci. Comput, vol.17, pp.1-4561, 2002. ,
Reliable real-time solution of parametrized partial differential equations: Reduced-Basis output bounds methods, Journal of Fluids Engineering, vol.124, issue.1, pp.70-80, 2002. ,
Reduced-Basis Methods for Partial Differential Equations, 2016. ,
DOI : 10.1007/978-3-319-15431-2
Primal Hybrid Finite Element Methods for 2nd Order Elliptic Equations, Mathematics of Computation, vol.31, issue.138, pp.31-391, 1977. ,
DOI : 10.2307/2006423
Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems. Part I, Computational Geosciences, vol.3, issue.3/4, pp.337-360, 1999. ,
DOI : 10.1023/A:1011591328604
Greedy algorithms for Reduced- Bases in Banach spaces, Constructive Approximation, vol.37, issue.3, pp.455-466, 2013. ,
Reduced-Basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations ? application to transport and continuum mechanics ,
DOI : 10.1007/s11831-008-9019-9
On the stability of the reduced basis method for Stokes equations in parametrized domains, Computer Methods in Applied Mechanics and Engineering, vol.196, issue.7, pp.1244-1260, 2007. ,
DOI : 10.1016/j.cma.2006.09.005
p-and hp¡FEM ? Theory and application to solid and fluid mechanics, 1998. ,
hp-FEM for Fluid Flow Simulation, pp.325-438, 1999. ,
DOI : 10.1007/978-3-662-03882-6_4
$hp$-Optimal discontinuous Galerkin methods for linear elliptic problems, Mathematics of Computation, vol.79, issue.272, pp.2117-2133, 2010. ,
DOI : 10.1090/S0025-5718-10-02335-5
URL : https://hal.archives-ouvertes.fr/hal-01090918
Singular integrals and differentiability properties of functions, Princeton Mathematical Series, issue.30, 1970. ,
On the Effect of the Internal Friction of Fluids on the Motion of Pendulums, Cambridge Philos. Trans, vol.9, pp.8-106, 1851. ,
DOI : 10.1017/CBO9780511702266.002
PolyMesher: a general-purpose mesh generator for polygonal elements written in Matlab. Structural and Multidisciplinary Optimization, pp.309-328, 2012. ,
hp-finite element discontinuous Galerkin approximations for the Stokes problem, M3AS, vol.12, issue.11, pp.1565-1616, 2002. ,
A Posteriori Error Estimates for Lowest-Order Mixed Finite Element Discretizations of Convection-Diffusion-Reaction Equations, SIAM Journal on Numerical Analysis, vol.45, issue.4 ,
DOI : 10.1137/060653184
MIXED FINITE ELEMENT METHODS: IMPLEMENTATION WITH ONE UNKNOWN PER ELEMENT, LOCAL FLUX EXPRESSIONS, POSITIVITY, POLYGONAL MESHES, AND RELATIONS TO OTHER METHODS, M3AS Math. Models Methods Appl ,
DOI : 10.1006/jcph.1998.6150
Hybridizable discontinuous Galerkin method (HDG) for Stokes interface flow, Journal of Computational Physics, vol.247, pp.262-278, 2013. ,
DOI : 10.1016/j.jcp.2013.03.064
HHOàHHOà desprobì emes issus de la mécanique des fluides. Nous considérons d'abord leprobì eme de Stokes, pour lequel nous obtenons une discrétisation d'ordre arbitraire inf-sup stable sur maillages généraux ,
Le schéma proposé permet de traiter des maillages généraux ainsi que de faire varier le degré polynomial d'unélémentàunélémentunélémentà l'autre. La dépendance de l'anisotropie locale du coefficient de diffusion est tracée explicitement dans l' ,
Méthodes Mixed High-Order, méthodes Hybrid High-Order, maillages généraux, analyse hp,probì eme d'Oseen,probì eme de Stokes,probì eme de Darcy ,